5128_Ch03_pp098-184

108 Chapter 3 Derivatives
Standardized Test Questions
You should solve the following problems without using a
graphing calculator.
36. True or False If f (x) x 2 x, then f (x) exists for every real
number x. Justify your answer. True. f(x) 2x 1
37. True or False If the left-hand derivative and the right-hand
derivative of f exist at x a, then f (a) exists. Justify your
answer. False. Let f(x) ⏐x⏐. The left hand derivative at x 0 is 1 and
the right hand derivative at x 0 is 1. f(0) does not exist.
38. Multiple Choice Let f (x) 4 3x. Which of the following
is equal to f (1)? C
(A) 7 (B) 7 (C) 3 (D) 3 (E) does not exist
39. Multiple Choice Let f (x) 1 3x 2 . Which of the following
is equal to f (1)? A
(A) 6 (B) 5 (C) 5 (D) 6 (E) does not exist
In Exercises 40 and 41, let
x 2 1, x 0
f x { 2x 1, x 0.
40. Multiple Choice Which of the following is equal to the lefthand
derivative of f at x 0? B
(A) 2 (B) 0 (C) 2 (D) (E)
41. Multiple Choice Which of the following is equal to the righthand
derivative of f at x 0? C
(A) 2 (B) 0 (C) 2 (D) (E)
Explorations
42.
x 2 , x 1
Let f x { 2x, x 1.
(a) Find f x for x 1. 2x (b) Find f x for x 1.
(c) Find lim x→1
f x. 2 (d) Find lim x→1
f x. 2
(e) Does lim x→1 f x exist? Explain. Yes, the two one-sided
limits exist and are the same.
(f) Use the definition to find the left-hand derivative of f
at x 1 if it exists. 2
(g) Use the definition to find the right-hand derivative of f
at x 1 if it exists. Does not exist
(h) Does f 1 exist? Explain. It does not exist because the righthand
derivative does not exist.
43. Group Activity Using graphing calculators, have each person
in your group do the following:
(a) pick two numbers a and b between 1 and 10;
(b) graph the function y x ax b;
(c) graph the derivative of your function (it will be a line with
slope 2);
(d) find the y-intercept of your derivative graph.
(e) Compare your answers and determine a simple way to predict
the y-intercept, given the values of a and b. Test your result.
The y-intercept is b a.
Extending the Ideas
44. Find the unique value of k that makes the function
x 3 , x 1
f x { 3x k, x 1
differentiable at x 1.
k 2
2
45. Generating the Birthday Probabilities Example 5 of this
section concerns the probability that, in a group of n people,
at least two people will share a common birthday. You can
generate these probabilities on your calculator for values of n
from 1 to 365.
Step 1: Set the values of N and P to zero:
0 N:0 P
Step 2: Type in this single, multi-step command:
N+1 N:1–(1–P) (366
–N)/365 P: {N,P}
Now each time you press the ENTER key, the command will
print a new value of N (the number of people in the room)
alongside P (the probability that at least two of them share a
common birthday):
0
{1 0}
{2 .002739726}
{3 .0082041659}
{4 .0163559125}
{5 .0271355737}
{6 .0404624836}
{7 .0562357031}
If you have some experience with probability, try to answer the
following questions without looking at the table:
(a) If there are three people in the room, what is the probability
that they all have different birthdays? (Assume that there are 365
possible birthdays, all of them equally likely.) 0.992
(b) If there are three people in the room, what is the probability
that at least two of them share a common birthday? 0.008
(c) Explain how you can use the answer in part (b) to find the
probability of a shared birthday when there are four people
in the room. (This is how the calculator statement in Step 2
generates the probabilities.)
(d) Is it reasonable to assume that all calendar dates are equally
likely birthdays? Explain your answer.
(c) If P is the answer to (b), then the probability of a shared birthday when
there are four people is
1 (1 P) 3 62
0.016.
365